Holomorphic Almost Modular Forms

Holomorphic almost modular forms are holomorphic functions of the complex upper half plane that can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in SL(2,Z). It is proved that such functions have a rotation‐invariant limit distribution when the argument approaches the real axis. An example of a holomorphic almost modular form is the logarithm of∏ n = 1 ∞( 1 − exp ( 2 π   i   n 2 z ) ). The paper is motivated by the author's previous studies [ Int. Math. Res. Not. 39 (2003) 2131–2151] on the connection between almost modular functions and the distribution of the sequence n 2 x modulo one. 2000 Mathematics Subject Classification 11F11 (primary), 11F06, 11J71 (secondary).